Question: When a number $n$ is increased by $50\%$ and the resulting number is divided by $3$, the result is $19$. What is the value of $n$?
Solution: Increasing by $50\%$ is the same as multiplying by $1.5$. So, $19 =\frac{1.5n}{3} =.5n$. Multiplying by $2$ shows that $n = \boxed{38}$.